A006073 -id:A006073 - cp's OEIS frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

2, 3, 4, 7, 8, 14, 16, 20, 21, 31, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 127, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175

Offset: 1

N. J. A. Sloane

Sequence is infinite, as proved by Schlage-Puchta, who comments: "Buttkewitz found a non-computational proof, and the Goldston-Pintz-Yildirim-sieve yields more precise information". - Charles R Greathouse IV, Jan 09 2013

The asymptotic density of this sequence is 0 (Erdős, 1936). - Amiram Eldar, Sep 17 2024

Calvin C. Clawson, Mathematical mysteries, Plenum Press, 1996, p. 250.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from T. D. Noe)
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.
Jan-Christoph Schlage-Puchta, The equation ω(n)=ω(n+1), Mathematika, Vol. 50, No. 1-2 (2003), pp. 99-101; arXiv preprint, arXiv:1105.1621 [math.NT], 2011.

Cf. A001221, A052215, A107800, A006073, A294277, A294278.

Subsequence of A062974.

import Data.List (elemIndices)
a006049 n = a006049_list !! (n-1)
a006049_list = map (+ 1) $ elemIndices 0 $
   zipWith (-) (tail a001221_list) a001221_list
-- Reinhard Zumkeller, Jan 22 2013

f[n_] := Length@FactorInteger[n];t = f /@ Range[175];Flatten@Position[Rest[t] - Most[t], 0] (* Ray Chandler, Mar 27 2007 *)
Select[Range[200],PrimeNu[#]==PrimeNu[#+1]&] (* Harvey P. Dale, May 09 2012 *)
Flatten[Position[Partition[PrimeNu[Range[200]],2,1],?(#[[1]]==#[[2]]&),{1},Heads->False]] (* _Harvey P. Dale, May 22 2015 *)
SequencePosition[PrimeNu[Range[200]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2019 *)

is(n)=omega(n)==omega(n+1) \\ Charles R Greathouse IV, Jan 09 2013

A001221(a(n)) = A001221(a(n)+1). - Reinhard Zumkeller, Jan 22 2013

Extended by Ray Chandler, Mar 27 2007

A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

Original entry on oeis.org

33, 85, 93, 121, 141, 170, 201, 213, 217, 244, 284, 301, 393, 428, 434, 445, 506, 602, 603, 604, 633, 637, 697, 841, 921, 962, 1041, 1074, 1083, 1084, 1130, 1137, 1244, 1261, 1274, 1309, 1345, 1401, 1412, 1430, 1434, 1448, 1490, 1532, 1556, 1586, 1604

Offset: 1

David W. Wilson

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), this sequence (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).
A056809 is a subsequence.
Cf. A006073. - Harvey P. Dale, Apr 19 2011

f[n_]:=Plus@@Last/@FactorInteger[n];lst={};lst={};Do[If[f[n]==f[n+1]==f[n+2],AppendTo[lst,n]],{n,0,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
pd2Q[n_]:=PrimeOmega[n]==PrimeOmega[n+1]==PrimeOmega[n+2]; Select[Range[1700],pd2Q]  (* Harvey P. Dale, Apr 19 2011 *)
SequencePosition[PrimeOmega[Range[1700]],{x_,x_,x_}][[;;,1]] (* Harvey P. Dale, Mar 08 2023 *)

is(n)=my(t=bigomega(n)); bigomega(n+1)==t && bigomega(n+2)==t \\ Charles R Greathouse IV, Sep 14 2015

list(lim)=my(v=List(),a=1,b=1,c); forfactored(n=4,lim\1+2,c=bigomega(n); if(a==b&&a==c, listput(v,n[1]-2)); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, May 07 2020

A055927 Numbers k such that k + 1 has one more divisor than k.

Original entry on oeis.org

1, 3, 9, 15, 25, 63, 121, 195, 255, 361, 483, 729, 841, 1443, 3363, 3481, 3721, 5041, 6241, 10201, 15625, 17161, 18224, 19321, 24963, 31683, 32761, 39601, 58564, 59049, 65535, 73441, 88208, 110889, 121801, 143641, 145923, 149769, 167281

Offset: 1

Labos Elemer, Jul 21 2000

nonn

Numbers k such that d(k+1) - d(k) = 1, where d(k) is A000005(k), the number of divisors.
Numbers k such that A049820(k) = A049820(k+1). - Jaroslav Krizek, Feb 10 2014
Numbers k such that A051950(k+1) = 1. - Danny Rorabaugh, Oct 05 2017

			a(4) = 15, as 15 has 4 and 16 has 5 divisors. a(6) = 63, as 63 and 64 have 6 and 7 divisors respectively.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Donovan Johnson)

Numbers where repetition occurs in A049820.

Cf. A000005, A006073, A045983, A049820, A075044.

Select[ Range[ 200000], DivisorSigma[0, # ] + 1 == DivisorSigma[0, # + 1] &]
Position[Differences[DivisorSigma[0,Range[170000]]],1]//Flatten (* Harvey P. Dale, Jul 06 2025 *)

for(n=1,1000,if(numdiv(n+1)-numdiv(n)==1,print1(n,", "))); /* Joerg Arndt, Apr 09 2011 */

More terms from David W. Wilson, Sep 06 2000, who remarks that every element is of form n^2 or n^2 - 1.

A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

Original entry on oeis.org

2, 33, 54, 55, 74, 85, 91, 92, 93, 115, 116, 133, 141, 142, 143, 144, 145, 158, 159, 175, 200, 205, 206, 212, 213, 214, 215, 216, 247, 295, 296, 301, 302, 323, 324, 325, 326, 332, 391, 392, 422, 445, 451, 535, 536, 542, 565, 632, 685, 686, 721, 722, 799, 800

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A006049, A006073, A045933-A045938.

Transpose[Select[Partition[Range[900],4,1],Length[Union[PrimeNu[#]]] == 1&]][[1]] (* Harvey P. Dale, Apr 12 2013 *)

A045938 Numbers n such that n through n+9 are divisible by the same number of distinct primes.

Original entry on oeis.org

48919, 184171, 218972, 218973, 320085, 320671, 343443, 353944, 397322, 403117, 435721, 492037, 526095, 526096, 526097, 526098, 526099, 534078, 534079, 534080, 583340, 607116, 636332, 693841, 701595, 761492, 822260, 919998, 942528

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A006049, A006073, A045932-A045937.

Offset corrected by Amiram Eldar, Oct 26 2019

A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

Original entry on oeis.org

1042404, 3460280, 3818828, 3998664, 4638984, 4991964, 5540248, 5701254, 5715500, 5964958, 6772050, 6794084, 7237384, 7453964, 7459088, 7745318, 7757034, 7993194, 8083634, 8153430, 8168194, 8273628, 8340834, 8340980, 8414756, 8486994, 8698898, 8722634, 8758904

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

			1042404 = 2^2*3*11*53*149, 1042405 = 5*6*143*29*79 and 1042406 = 2*17*23*31*43 each have 5 distinct prime factors, so 1042404 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A192203 (subsequence for squarefree triples). Subsequence of A140079 (2 consec.) and of A006073.

Cf. A364308 (3 dist. factors), A364309 (4 dist. factors), A364265 (6 dist. factors), A001221, A087978.

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 5 then
        if omega(k+1) = 5 then
            if omega(k+2) = 5 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

seq[lim_] := Module[{s  = {}, q1 = False, q2 = False, q3}, Do[q3 = PrimeNu[k] == 5; If[q1 && q2 && q3, AppendTo[s, k-2]]; q1 = q2; q2 = q3, {k, 3, lim}]; s]; seq[10^7] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A087978(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 5}. - R. J. Mathar, Jul 18 2023

A045933 Numbers n such that n through n+4 are divisible by the same number of distinct primes.

Original entry on oeis.org

54, 91, 92, 115, 141, 142, 143, 144, 158, 205, 212, 213, 214, 215, 295, 301, 323, 324, 325, 391, 535, 685, 721, 799, 1135, 1345, 1465, 1535, 1711, 1941, 1981, 2101, 2215, 2302, 2303, 2304, 2425, 2641, 2664, 2714, 3865, 3912, 4411, 5450, 5461, 6354, 6505

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A006049, A006073, A045932, A045934-A045938.

SequencePosition[PrimeNu[Range[7000]],{x_,x_,x_,x_,x_}][[All,1]] (* Harvey P. Dale, Jun 13 2022 *)

A338453 Starts of runs of 3 consecutive numbers with the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 242, 243, 1837, 2361, 3693, 3728, 6061, 6457, 9782, 11181, 11721, 13855, 15177, 20017, 22591, 28021, 31461, 31887, 33098, 33993, 38137, 52016, 52112, 60321, 76897, 78542, 78745, 80461, 108394, 116017, 119541, 124453, 125493, 127117, 127417, 145369, 151805, 154113

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1) = A093653(k+2).

			3 is a term since A093653(3) = A093653(4) = A093653(5) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A093653.
Subsequence of A338452.
Similar sequences: A005238, A006073, A045939.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; m = 3; fs = f /@ Range[m]; Do[If[Equal @@  fs, AppendTo[s, n - m]]; fs = Rest @ AppendTo[fs, f[n]], {n, m + 1, 155000}]; s
SequencePosition[Table[Total[DigitCount[Divisors[n],2,1]],{n,160000}],{x_,x_,x_}][[All,1]] (* Harvey P. Dale, Feb 04 2023 *)

A364307 Numbers k such that k, k+1 and k+2 have exactly 2 distinct prime factors.

Original entry on oeis.org

20, 33, 34, 38, 44, 50, 54, 55, 56, 74, 75, 85, 86, 91, 92, 93, 94, 98, 115, 116, 117, 122, 133, 134, 141, 142, 143, 144, 145, 146, 158, 159, 160, 175, 176, 183, 187, 200, 201, 205, 206, 207, 212, 213, 214, 215, 216, 217, 224, 235, 247, 248, 295, 296

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			44 = 2^2*11 has 2 distinct prime factors, and so has 45 = 3^2*5 and so has 46 = 2*23, so 44 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A074851.
Cf. A364308 (3 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221.
A039833 is a subsequence.

q[n_] := q[n] = PrimeNu[n] == 2; Select[Range[300], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 2}.

A364308 Numbers k such that k, k+1 and k+2 have exactly 3 distinct prime factors.

Original entry on oeis.org

644, 740, 804, 986, 1034, 1064, 1104, 1220, 1274, 1308, 1309, 1462, 1494, 1580, 1748, 1884, 1885, 1924, 1988, 2013, 2014, 2108, 2134, 2254, 2288, 2294, 2330, 2354, 2364, 2408, 2464, 2484, 2540, 2583, 2584, 2664, 2665, 2666, 2678, 2684, 2714, 2715, 2716, 2754, 2793

Offset: 1

R. J. Mathar, Jul 18 2023

nonn

			644 = 2^2*7*23 has 3 distinct prime factors, 645 = 3*5*43 has 3 distinct prime factors, and 646 = 2*17*19 has 3 distinct prime factors, so 644 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A006073 and of A140077.

Cf. A364307 (2 factors), A364309 (4 factors), A364266 (5 factors), A364265 (6 factors), A001221, A080569.

q[n_] := q[n] = PrimeNu[n] == 3; Select[Range[3000], q[#] && q[#+1] && q[#+2] &] (* Amiram Eldar, Oct 01 2024 *)

a(1) = A080569(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 3}.

cp's OEIS Frontend

A006049 Numbers k such that k and k+1 have the same number of distinct prime divisors.

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A045939 Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities).

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A055927 Numbers k such that k + 1 has one more divisor than k.

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A045932 Numbers n such that n through n+3 are divisible by the same number of distinct primes.

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A045938 Numbers n such that n through n+9 are divisible by the same number of distinct primes.

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A364266 The first term in a chain of at least 3 consecutive numbers each with exactly 5 distinct prime factors.

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A045933 Numbers n such that n through n+4 are divisible by the same number of distinct primes.

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A338453 Starts of runs of 3 consecutive numbers with the same total binary weight of their divisors (A093653).

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